st: Endogeneity and instrument selection

Hello,
I am estimating an IV-FE model and puzzled with the results on the
endogeneity test of the possibly endogenous covariates. I would like
to summarize my model and for suggestions.
Let us forget about the subscripts i and t defining individuals and
time. My model is as follows:
H = A(1) * X + A(2) * W1 + A(3) * W2 + U (Eq 1)
I believe that W1 and W2 is endogenous and hence instrument it with
the variables E1, E2, E3, and E4 as follows (for W1):
W1 = B(1) * X + B(2) * E1 + B(3) * E2 + B(4) * E3 + B(5) * E4 + V (Eq 2)
I find that E1, E2, E3, and E4 are in some cases individually but
besides jointly "relevant" (with t or an F test) and "exogenous" (with
respect to the overidentification test). Hence, the instruments are
valid.
I estimate the model with E1, E2, E3, and E4 as set above, and when I
test to see if W is indeed "endogenous" I find that it is endogenous
using the following command:
xtivreg2 H (W1 W2 = E1 E2 E3 E4) X, fe robust cluster(HHIDPN) endog(W1 W2)
Then I exclude E4 and I find that W is NOT endogenous anymore. This is
what puzzles me and I cannot think of any reason why it would be the
case.
In particular, if I am not mistaken, the endogeneity test works as
follows (simplified for W1):
W1 can be correlated with U only if V is correlated with U because we
know that "X, E1, E2, E3, E4" are uncorrelated with U. And then we
test if V is significant in Equation 1.
Given this, I cannot really think of any reason why W1 becomes
exogenous when I drop E4 from the regression.
I would appreciate your thoughts.
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